Average Error: 0.5 → 0.9
Time: 6.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v + \sqrt{1}}}{v - \sqrt{1}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v + \sqrt{1}}}{v - \sqrt{1}}\right)
double f(double v) {
        double r265750 = 1.0;
        double r265751 = 5.0;
        double r265752 = v;
        double r265753 = r265752 * r265752;
        double r265754 = r265751 * r265753;
        double r265755 = r265750 - r265754;
        double r265756 = r265753 - r265750;
        double r265757 = r265755 / r265756;
        double r265758 = acos(r265757);
        return r265758;
}
↓
double f(double v) {
        double r265759 = atan2(1.0, 0.0);
        double r265760 = 2.0;
        double r265761 = r265759 / r265760;
        double r265762 = 1.0;
        double r265763 = 5.0;
        double r265764 = v;
        double r265765 = r265764 * r265764;
        double r265766 = r265763 * r265765;
        double r265767 = r265762 - r265766;
        double r265768 = sqrt(r265762);
        double r265769 = r265764 + r265768;
        double r265770 = r265767 / r265769;
        double r265771 = r265764 - r265768;
        double r265772 = r265770 / r265771;
        double r265773 = asin(r265772);
        double r265774 = r265761 - r265773;
        return r265774;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied acos-asin0.5

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}\right)\]

  6. Applied difference-of-squares0.9

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(v + \sqrt{1}\right) \cdot \left(v - \sqrt{1}\right)}}\right)\]

  7. Applied associate-/r*0.9

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \color{blue}{\left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v + \sqrt{1}}}{v - \sqrt{1}}\right)}\]

  8. Final simplification0.9

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v + \sqrt{1}}}{v - \sqrt{1}}\right)\]

Reproduce

herbie shell --seed 2020083 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))